ON THE DIAMETER OF INTERSECTION GRAPHS OF FINITE GROUPS

Abstract The intersection graph Δ G $\Delta _G$ normal upper Delta Subscript upper G of a finite group G $G$ upper G is a simple graph with vertices the nontrivial proper subgroups of G $G$ upper G , and an edge between two vertices if their corresponding subgroups intersect nontrivially. These graphs were introduced by Csákány and Pollák [‘The graph of subgroups of a finite group’, Czechoslovak Math. J. 19 (94) (1969), 241–247]. We consider two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a nonsimple group to have an intersection graph of diameter 4. We also construct the first examples of alternating groups whose intersection graphs have diameter 4. In fact, we show that there is an infinite family of such groups.